Cos2phi Qubit¶

Without disorder in circuit parameters, the cos2phi qubit [Smith2020] is described by the Hamiltonian

\[\begin{split}H = & \,2 E_\text{CJ}n_\phi^2 + 2 E_\text{CJ} (n_\theta - n_\text{g} - n_\zeta)^2 + 4 E_\text{C} n_\zeta^2\\ & + E_\text{L}(\phi - \pi\Phi_\text{ext}/\Phi_0)^2 + E_\text{L} \zeta^2 - 2E_\text{J}\cos{\theta}\cos{\phi}.\end{split}\]

In the presence of disorder, the circuit is described by

\[\begin{split}H = & \,2 E_\text{CJ}'n_\phi^2 + 2 E_\text{CJ}' (n_\theta - n_\text{g} - n_\zeta)^2 + 4 E_\text{C} n_\zeta^2\\ & + E_\text{L}'(\phi - \pi\Phi_\text{ext}/\Phi_0)^2 + E_\text{L}' \zeta^2 - 2 E_\text{J}\cos{\theta}\cos{\phi} \\ & + 2 dE_\text{J} E_\text{J}\sin{\theta}\sin{\phi} \\ & - 4 dC_\text{J} E_\text{CJ}' n_\phi (n_\theta - n_\text{g}-n_\zeta) \\ & + dL E_\text{L}'(2\phi - \varphi_\text{ext})\zeta ,\end{split}\]

where \(E_\text{CJ}' = E_\text{CJ} / (1 - dC_\text{J})^2\) and \(E_\text{L}' = E_\text{L} / (1 - dL)^2\).

Here, the disorder is defined as follows: the inductive energies of the two inductors are \(E_\text{L1,2} = E_\text{L}/(1 \pm dL)\); the charging energies of the two Josephson junctions are \(E_\text{CJ1,2} = E_\text{CJ}/(1 \pm dC_\text{J})\); the junction energies of the two Josephson junctions are \(E_\text{J1,2} = E_\text{J} (1 \pm dE_\text{J})\). Alternatively, the above relations can be rewritten as: \(dL = (E_\text{L2}-E_\text{L1})/(E_\text{L2}+E_\text{L1}), E_\text{L} = 2E_\text{L1}E_\text{L2}/(E_\text{L1}+E_\text{L2})\) for inductive energies, \(dC_\text{J} = (E_\text{CJ2}-E_\text{CJ1})/(E_\text{CJ1}+E_\text{CJ2}), E_\text{CJ} = 2E_\text{CJ1}E_\text{CJ2}/(E_\text{CJ1}+E_\text{CJ2})\) for charging energies, and \(dE_\text{J} = (E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2}), E_\text{J} = (E_\text{J1}+E_\text{J2})/2\) for junction energies.

Here, we adopt notation that is consistent with other qubit classes. A conversion to the notation used in Ref. [Smith2020] can be found in the following table.

Notation used here	Notation used in Ref. [Smith2020]
\(\zeta\)	\(\theta\)
\(\theta\)	\(\varphi\)
\(\phi\)	\(\phi/2\)
\(E_\text{C}\)	\(E_\text{C} x\)
\(E_\text{CJ}\)	\(E_\text{C}\)

To numerically diagonalize the Hamiltonian of the cos2phi qubit, the harmonic basis is used for both the \(\phi\) and \(\zeta\) variables, and the charge basis is used for the \(\theta\) variable. The user needs to specify cutoffs for basis states described above, i.e., phi_cut, zeta_cut, and ncut, chosen large enough so that convergence is achieved.

An instance of the cos2phi qubit is initialized as follows:

cos2phi_qubit = scqubits.Cos2PhiQubit(EJ = 15.0,
                                      ECJ = 2.0,
                                      EL = 1.0,
                                      EC = 0.04,
                                      dCJ = 0.0,
                                      dL = 0.6,
                                      dEJ = 0.0,
                                      flux = 0.5,
                                      ng = 0.0,
                                      ncut = 7,
                                      phi_cut = 7,
                                      zeta_cut = 30)

From within Jupyter notebook, a cos2phi qubit instance can alternatively be created with:

cos2phi_qubit = scqubits.Cosi2PhiQubit.create()

This functionality is enabled if the ipywidgets package is installed, and displays GUI forms prompting for the entry of the required parameters.

Cos2phi Qubit¶

Wavefunctions and visualization of eigenstates and the potential¶

Implemented operators¶

Computation and visualization of matrix elements¶

Estimation of coherence times¶

Cos2phi Qubit¶

Calculational methods related to Hamiltonian and energy spectra¶

Wavefunctions and visualization of eigenstates and the potential¶

Implemented operators¶

Computation and visualization of matrix elements¶

Estimation of coherence times¶